Steffen's polyhedron

In geometry, Steffen's polyhedron is a flexible polyhedron discovered (in 1978[1]) by and named after Klaus Steffen. It is based on the Bricard octahedron, but unlike the Bricard octahedron its surface does not cross itself.[2] With nine vertices, 21 edges, and 14 triangular faces, it is the simplest possible non-crossing flexible polyhedron.[3] Its faces can be decomposed into three subsets: two six-triangle-patches from a Bricard octahedron, and two more triangles (the central two triangles of the net shown in the illustration) that link these patches together.[4]

Steffen's polyhedron
A net for Steffen's polyhedron. The solid and dashed lines represent mountain folds and valley folds, respectively.

It obeys the strong bellows conjecture, meaning that (like the Bricard octahedron on which it is based) its Dehn invariant stays constant as it flexes.[5]

References
  1. Optimizing the Steffen flexible polyhedron Lijingjiao et al. 2015
  2. Connelly, Robert (1981), "Flexing surfaces", in Klarner, David A. (ed.), The Mathematical Gardner, Springer, pp. 79–89, doi:10.1007/978-1-4684-6686-7_10, ISBN 978-1-4684-6688-1.
  3. Demaine, Erik D.; O'Rourke, Joseph (2007), "23.2 Flexible polyhedra", Geometric Folding Algorithms: Linkages, origami, polyhedra, Cambridge University Press, Cambridge, pp. 345–348, doi:10.1017/CBO9780511735172, ISBN 978-0-521-85757-4, MR 2354878.
  4. Fuchs, Dmitry; Tabachnikov, Serge (2007), Mathematical Omnibus: Thirty lectures on classic mathematics, Providence, RI: American Mathematical Society, p. 354, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, MR 2350979.
  5. Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", Journal of Geometry, 99 (1–2): 1–13, arXiv:0901.2989, doi:10.1007/s00022-011-0061-7, MR 2823098.
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